#231232

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(2,5)(6,8)(10,13)(11,15)(14,16)(17,18,21,25,22,26,30,24)(19,23,28,32,27,31,20,29);(1,2)(3,6)(4,5)(7,8)(10,14)(13,16)(17,19,21,28,22,27,30,20)(18,23,25,32,26,31,24,29);(1,3,4,7)(2,6,5,8)(9,10,15,16,12,14,11,13)(17,20,30,27,22,28,21,19)(18,23,25,32,26,31,24,29);(2,5)(6,8)(19,27)(20,28)(23,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=efh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=chi,d^b=dei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138428);